A Simple Model of Concentrated Nonlinearity

Adami, Riccardo; Teta, Alessandro

doi:10.1007/978-3-0348-8745-8_13

Cited by 15 publications

(18 citation statements)

References 6 publications

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“…A detailed analytical study of the non-linear case σ > 0 (which is no longer explicitly solvable) can be found in ( [13,11]). The authors obtained general results about existence of solutions either local or global in time and proved existence of blow up solutions for σ ≥ 1.…”

Section: Nonlinear Point Interactionsmentioning

confidence: 99%

The quantum beating and its numerical simulation

Carlone

Figari

Negulescu

2017

Journal of Mathematical Analysis and Applications

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Abstract. We examine the suppression of quantum beating in a one dimensional nonlinear double well potential, made up of two focusing nonlinear point interactions. The investigation of the Schrödinger dynamics is reduced to the study of a system of coupled nonlinear Volterra integral equations. For various values of the geometric and dynamical parameters of the model we give analytical and numerical results on the way states, which are initially confined in one well, evolve. We show that already for a nonlinearity exponent well below the critical value there is complete suppression of the typical beating behavior characterizing the linear quantum case.

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Section: Nonlinear Point Interactionsmentioning

confidence: 99%

The quantum beating and its numerical simulation

Carlone

Figari

Negulescu

2017

Journal of Mathematical Analysis and Applications

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“…In this note we introduce a model of nonlinear Schrödinger equation in R 3 with a nonlinearity concentrated in a finite number of points. The corresponding case in dimension one was studied in [1,2]. The three-dimensional case exhibits a more singular character.…”

Section: Introductionmentioning

confidence: 99%

“…-In the case n = 1, σ1 = σ = 1, γ 1 = γ < 0, if ψ 0 L 2 (R 3 ) < (4π √ 2|γ |) −1 , then the solution ψ(t) is global in time.Proof. -Without loss of generality, we choose y 1 = 0 and define:ξ(t, x) = |x| ψ(t) m |x| = |x| φ(t) m |x| + q(t) 4π , x ∈ R, t ∈ [0, T * ).…”

mentioning

confidence: 99%

The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity

Adami

Dell’Antonio

Figari

et al. 2003

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider the Schrödinger equation in R 3 with nonlinearity concentrated in a finite set of points. We formulate the problem in the space of finite energy V , which is strictly larger than the standard H 1 -space due to the specific singularity exhibited by the solutions. We prove local existence and, for a repulsive or weakly attractive nonlinearity, also global existence of the solutions.  2003 Éditions scientifiques et médicales Elsevier SAS Keywords: Nonlinear Schrödinger Equation (NLSE); Point interactions; Nonlinear Dirac Delta potentials; Existence and uniqueness in energy space RÉSUMÉ. -On considère l'équation de Schrödinger avec une nonlinéarité concentrée en un nombre fini de points. On formule le problème dans l'espace V d'énergie finie, qui contient strictement l'espace standard H 1 , à cause de la singularité spécifique des solutions. On prouve des résultats d'existence locale et même, pour une nonlinéarité répulsive ou faiblement attractive, l'existence globale des solutions.  2003 Éditions scientifiques et médicales Elsevier SAS

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“…We shall recall the definition of the equation and the basic results about its well-posedness, this part is a survey of the works [4,5] and [16]. Then we shall state the main result of [10] on the derivation of the equation as the limit of Schrödinger equations with spatially non-homogeneous scaled nonlinearity, and discuss the basic ideas behind the proof.…”

Section: Nonlinear Delta-interactions In Dimension Onementioning

confidence: 99%

“…We shall follow [4,5] and consider only power-type nonlinearities, i.e., α j (z) = γ j z µ j for some real constants γ j , and µ j > 0, but more general nonlinearities could be considered, see, e.g., [16]. Our model nonlinear delta-interaction is then defined by the integral equation…”

Section: Consider the Cauchy Problemmentioning

confidence: 99%

On the derivation of the Schrödinger equation with point-like nonlinearity

Cacciapuoti

2015

Nanosist.: fiz. him. mat.

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In this report we discuss the problem of approximating nonlinear delta-interactions in dimensions one and three with regular, local or non-local nonlinearities. Concerning the one dimensional case, we discuss a recent result proved in [10], on the derivation of nonlinear delta-interactions as limit of scaled, local nonlinearities. For the three dimensional case, we consider an equation with scaled, non-local nonlinearity. We conjecture that such an equation approximates the nonlinear delta-interaction, and give an heuristic argument to support our conjecture.

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A Simple Model of Concentrated Nonlinearity

Cited by 15 publications

References 6 publications

The quantum beating and its numerical simulation

The quantum beating and its numerical simulation

The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity

On the derivation of the Schrödinger equation with point-like nonlinearity

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